DESIGN AND SIMULATION OF PHOTOVOLTAIC WATER PUMPING SYSTEM

DESIGN AND SIMULATION OF PHOTOVOLTAIC WATER PUMPING SYSTEM A Thesis Presented to the Faculty of California Polytechnic State University, San Luis Obispo In Partial Fulfillment of the Requirements for the Degree of Master of Science in Electrical Engineering by Akihiro Oi September 2005

AUTHORIZATION FOR REPRODUCTION OF MASTER S THESIS I grant permission for the reproduction of this thesis in its entirety or any of its parts, without further authorization from me. Signature (Akihiro Oi) Date ii

APPROVAL Title: DESIGN AND SIMULATION OF PHOTOVOLTAIC WATER PUMPING SYSTEM Author: Akihiro Oi Date Submitted: 26 th September, 2005 Dr. Taufik Committee Chair Signature Dr. Ahmad Nafisi Committee Member Signature Dr. William Ahlgren Committee Member Signature iii

ABSTRACT DESIGN AND SIMULATION OF PHOTOVOLTAIC WATER PUMPING SYSTEM Akihiro Oi This thesis deals with the design and simulation of a simple but efficient photovoltaic water pumping system. It provides theoretical studies of photovoltaics and modeling techniques using equivalent electric circuits. The system employs the maximum power point tracker (MPPT). The investigation includes discussion of various MPPT algorithms and control methods. PSpice simulations verify the DC-DC converter design. MATLAB simulations perform comparative tests of two popular MPPT algorithms using actual irradiance data. The thesis decides on the output sensing direct control method because it requires fewer sensors. This allows a lower cost system. Each subsystem is modeled in order to simulate the whole system in MATLAB. It employs SIMULINK to model a DC pump motor, and the model is transferred into MATLAB. Then, MATLAB simulations verify the system and functionality of MPPT. Simulations also make comparisons with the system without MPPT in terms of total energy produced and total volume of water pumped per day. The results validate that MPPT can significantly increase the efficiency and the performance of PV water pumping system compared to the system without MPPT. iv

ACKNOWLEDGEMENTS I would like to first acknowledge my advisor, Dr. Taufik, for his support and advice throughout my graduate program. His power electronics courses and his dedication to his students gave me the best experience during the program. I would also like to express my sincere appreciation to my other thesis committees, Dr. Nafisi and Dr. Ahlgren, for review of this thesis in detail and their important feedback. I would like to thank my colleague and fri, John Carlin, who has a career experience in designing photovoltaic systems. A number of ideas generated from our numerous discussions and his feedback are incorporated in this thesis. Also, thanks to my other good colleagues, James Silva, John Cadwell, Michael Chong, James Sorenson, Sajiv Nair, Alan Yeung, Yat Tam, all other denizens of EE Grad Lab and the lab technicians for their support and willingness to help me out during various stages of my project. Finally, to my parents, my sister, and my fris - many thanks for much support the whole way through, especially Jenny Ho for her constant encouragement and support during two years of my graduate work. Akihiro Oi September 2005 v

Table of Contents List of Tables... viii List of Figures...ix Chapter 1 Introduction...1 1.1 Water Pumping Systems and Photovoltaic Power... 1 1.2 Energy Storage Alternatives... 3 1.3 The Proposed System... 4 1.3.1 PV Module...5 1.3.2 Maximum Power Point Tracker...5 1.3.3 MPPT Controller...6 1.3.4 Water Pump...7 1.4 Background and Scope of This Thesis... 8 Chapter 2 Photovoltaic Modules...10 2.1 Introduction... 10 2.2 Photovoltaic Cell... 10 2.3 Modeling a PV Cell... 12 2.3.1 The Simplest Model...12 2.3.2 The More Accurate Model...15 2.4 Photovoltaic Module... 17 2.5 Modeling a PV Module by MATLAB... 18 2.6 The I-V Curve and Maximum Power Point... 25 Chapter 3 Maximum Power Point Tracker...27 3.1 Introduction... 27 3.2 I-V Characteristics of DC Motors... 28 3.3 DC-DC Converter... 31 3.3.1 Topologies...31 3.3.2 Cúk and SEPIC Converters...32 3.3.3 Basic Operation of Cúk Converter...34 3.4 Mechanism of Load Matching... 37 3.5 Maximum Power Point Tracking Algorithms... 38 3.5.1 Perturb & Observe Algorithm...40 3.5.2 Incremental Conductance Algorithm...44 3.6 Control of MPPT... 47 3.6.1 PI Control...47 3.6.2 Direct Control...48 3.6.3 Output Sensing Direct Control...50 3.7 Limitations of MPPT... 52 Chapter 4 Design and Simulations...55 4.1 Introduction... 55 4.2 Cúk Converter Design... 55 4.2.1 Component Selection...56 4.2.2 PSpice Simulations...59 4.2.3 Choice of MPPT Sampling Rate...61 4.3 Comparisons of P&O and inccond Algorithm... 62 4.4 MPPT Simulations with Resistive Load... 66 vi

4.5 MPPT Simulations with DC Pump Motor Load... 70 4.5.1 Modeling of DC Water Pump...71 4.5.2 MATLAB Simulation Results...73 4.6 System with MPPT vs. Direct-coupled System... 75 Chapter 5 Conclusion...78 5.1 Summary... 78 5.2 Difficulties and Future Research... 79 5.3 Concluding Remarks... 80 Bibliography...81 Appix A...84 A.1 MATLAB Functions and Scripts... 84 A.1.1 MATLAB Function for Modeling BP SX 150S PV Module...84 A.1.2 MATLAB Script to Draw PV I-V Curves...85 A.1.3 MATLAB Function to Find the MPP...86 A.1.4 MATLAB Script: P&O Algorithm...86 A.1.5 MATLAB Script: inccond Algorithm...88 A.1.6 MATLAB Script for MPPT with Output Sensing Direct Control Method...90 A.1.7 MATLAB Script for MPPT Simulations with DC Pump Motor Load...93 A.1.8 MATLAB Script for MPPT Simulations with Direct-coupled DC Water Pump...97 A.2 MPPT Simulations with Resistive Load... 100 A.2.1 Direct Control Method with P&O Algorithm...100 A.2.2 Direct Control Method with inccond Algorithm...101 Appix B...102 B.1 DSP Control... 102 B.1.1 TMS320F2812 DSP...102 B.1.2 SIMULNK and TI DSP...102 B.1.3 Example...103 vii

List of Tables Table 1-1: PV powered, Diesel powered, vs. Windmill [13]... 3 Table 2-1: Electrical characteristics data of PV module taken from the datasheet [1]... 18 Table 3-1: Load matching with the resistive load (6) under the varying irradiance... 53 Table 3-2: Load matching with the resistive load (12) under the varying irradiance... 53 Table 4-1: Design specification of the Cúk Converter... 55 Table 4-2: Cúk converter design: comparisons of simulations and calculated results... 60 Table 4-3: Comparison of the P&O and inccond algorithms on a cloudy day... 65 Table 4-4: Energy production and efficiency of PV module with and without MPPT... 75 Table 4-5: Total volume of water pumped for 12 hours... 77 viii

List of Figures Figure 1-1: Block diagram of the proposed PV water pumping system... 5 Figure 2-1: Illustration of the p-n junction of PV cell [16]... 11 Figure 2-2: Illustrated side view of solar cell and the conducting current [16]... 11 Figure 2-3: PV cell with a load and its simple equivalent circuit [16]... 12 Figure 2-4: Diagrams showing a short-circuit and an open-circuit condition [16]... 13 Figure 2-5: I-V plot of ideal PV cell under two different levels of irradiance (25 o C)... 15 Figure 2-6: More accurate equivalent circuit of PV cell... 16 Figure 2-7: PV cells are connected in series to make up a PV module... 17 Figure 2-8: Picture of BP SX 150S PV module [1]... 18 Figure 2-9: Equivalent circuit used in the MATLAB simulations... 19 Figure 2-10: Effect of diode ideally factors by MATLAB simulation (1KW/m 2, 25 o C)... 21 Figure 2-11: Effect of series resistances by MATLAB simulation (1KW/m 2, 25 o C)... 22 Figure 2-12: I-V curves of BP SX 150S PV module at various temperatures... 24 Figure 2-13: Simulated I-V curve of BP SX 150S PV module (1KW/m 2, 25 o C)... 25 Figure 2-14: I-V and P-V relationships of BP SX 150S PV module... 26 Figure 3-1: PV module is directly connected to a (variable) resistive load... 27 Figure 3-2: I-V curves of BP SX 150S PV module and various resistive loads... 28 Figure 3-3: Electrical model of permanent magnet DC motor... 29 Figure 3-4: PV I-V curves with varying irradiance and a DC motor I-V curve... 30 Figure 3-5: PV I-V curves with iso-power lines (dotted) and a DC motor I-V curve... 31 Figure 3-6: Circuit diagram of the basic Cúk converter... 34 Figure 3-7: Circuit diagram of the basic SEPIC converter... 34 Figure 3-8: Basic Cúk converter when the switch is ON... 35 Figure 3-9: Basic Cúk converter when the switch is OFF... 35 Figure 3-10: The impedance seen by PV is R in that is adjustable by duty cycle (D)... 38 Figure 3-11: I-V curves for varying irradiance and a trace of MPPs (25 o C)... 39 Figure 3-12: I-V curves for varying irradiance and a trace of MPPs (50 o C)... 40 Figure 3-13: Plot of power vs. voltage for BP SX 150S PV module (1KW/m 2, 25 o C)... 41 Figure 3-14: Flowchart of the P&O algorithm... 41 Figure 3-15: Erratic behavior of the P&O algorithm under rapidly increasing irradiance... 43 Figure 3-16: Flowchart of the inccond algorithm... 46 Figure 3-17: Block diagram of MPPT with the PI compensator... 48 Figure 3-18: Block diagram of MPPT with the direct control... 48 Figure 3-19: Relationship of the input impedance of Cúk converter and its duty cycle... 49 Figure 3-20: Output power of Cúk converter vs. its duty cycle (1KW/m 2, 25 o C)... 51 Figure 3-21: Flowchart of P&O algorithm for the output sensing direct control method... 52 Figure 4-1: Schematic of the Cúk converter with PMDC motor load... 59 Figure 4-2: PSpice plots of input/output current (above) and voltage (below)... 60 Figure 4-3: Transient response when duty cycle is increased 0.35% at 250ms... 61 Figure 4-4: Searching the MPP (1KW/m 2, 25 o C)... 62 Figure 4-5: Irradiance data for a sunny and a cloudy day of April in Barcelona, Spain [2]... 63 Figure 4-6: Traces of MPP tracking on a sunny day (25 o C)... 64 Figure 4-7: Trace of MPP tracking on a cloudy day (25 o C)... 65 ix

Figure 4-8: MPPT simulation flowchart... 68 Figure 4-9: MPPT simulations with the resistive load (100 to 1000W/m 2, 25 o C)... 69 Figure 4-10: Output protection & regulation (100 to 1000W/m 2, 25 o C)... 70 Figure 4-11: Kyocera SD 12-30 water pump performance chart [13]... 71 Figure 4-12: SIMULINK model of permanent magnet DC pump motor... 72 Figure 4-13: SIMULINK DC machine block parameters... 72 Figure 4-14: SIMULINK plot of R load ()... 73 Figure 4-15: MPPT simulations with the DC pump motor load (20 to 1000W/m 2, 25 o C)... 74 Figure 4-16: SIMULINK plot of DC motor I-V curve... 75 Figure 4-17: Flow rates of PV water pumps for a 12-hour period... 76 Figure A-1: MPPT Simulations with the direct control method (P&O algorithm)... 100 Figure A-2: MPPT Simulations with the direct control method (inccond algorithm)... 101 Figure B-1: A simple example of generating PWM from the voltage input... 103 Figure B-2: Plots of the input voltage and the PWM output shown as duty cycle... 103 x

Chapter 1 Introduction Water resources are essential for satisfying human needs, protecting health, and ensuring food production, energy and the restoration of ecosystems, as well as for social and economic development and for sustainable development [25]. However, according to UN World Water Development Report in 2003, it has been estimated that two billion people are affected by water shortages in over forty countries, and 1.1 billion do not have sufficient drinking water [26]. There is a great and urgent need to supply environmentally sound technology for the provision of drinking water. Remote water pumping systems are a key component in meeting this need. It will also be the first stage of the purification and desalination plants to produce potable water. In this thesis, a simple but efficient photovoltaic water pumping system is presented. It provides theoretical studies of photovoltaics (PV) and its modeling techniques. It also investigates in detail the maximum power point tracker (MPPT), a power electronic device that significantly increases the system efficiency. At last, it presents MATLAB simulations of the system and makes comparisons with a system without MPPT. 1.1 Water Pumping Systems and Photovoltaic Power A water pumping system needs a source of power to operate. In general, AC powered system is economic and takes minimum maintenance when AC power is available from the nearby power grid. However, in many rural areas, water sources are spread over many miles of land and power lines are scarce. Installation of a new transmission line and a transformer to the location is often prohibitively expensive. Windmills have been installed 1

traditionally in such areas; many of them are, however, inoperative now due to lack of proper maintenance and age. Today, many stand-alone type water pumping systems use internal combustion engines. These systems are portable and easy to install. However, they have some major disadvantages, such as: they require frequent site visits for refueling and maintenance, and furthermore diesel fuel is often expensive and not readily available in rural areas of many developing countries. The consumption of fossil fuels also has an environmental impact, in particular the release of carbon dioxide (CO 2 ) into the atmosphere. CO 2 emissions can be greatly reduced through the application of renewable energy technologies, which are already cost competitive with fossil fuels in many situations. Good examples include large-scale grid-connected wind turbines, solar water heating, and off-grid stand-alone PV systems [24]. The use of renewable energy for water pumping systems is, therefore, a very attractive proposition. Windmills are a long-established method of using renewable energy; however they are quickly phasing out from the scene despite success of large-scale grid-tied wind turbines. PV systems are highly reliable and are often chosen because they offer the lowest life-cycle cost, especially for applications requiring less than 10KW, where grid electricity is not available and where internal-combustion engines are expensive to operate [24]. If the water source is 1/3 mile (app. 0.53Km) or more from the power line, PV is a favorable economic choice [13]. Table 1-1 shows the comparisons of different stand-alone type water pumping systems. 2

System Type Advantages Disadvantages PV Powered System Low maintenance Unatted operation Reliable long life No fuel and no fumes Easy to install Low recurrent costs System is modular and closely matched to need Diesel (or Gas) Powered System Windmill Moderate capital costs Easy to install Can be portable Extensive experience available No fuel and no fumes Potentially long-lasting Works well in windy sites Table 1-1: PV powered, Diesel powered, vs. Windmill [13] Relatively high initial cost Low output in cloudy weather Needs maintenance and replacement Site visits necessary Noise, fume, dirt problems Fuel often expensive and supply intermittent High maintenance Seasonal disadvantages Difficult find parts thus costly repair Installation is labor intensive and needs special tools 1.2 Energy Storage Alternatives Needless to say, photovoltaics are able to produce electricity only when the sunlight is available, therefore stand-alone systems obviously need some sort of backup energy storage which makes them available through the night or bad weather conditions. Among many possible storage technologies, the lead-acid battery continues to be the workhorse of many PV systems because it is relatively inexpensive and widely available. In addition to energy storage, the battery also has ability to provide surges of current that are much higher than the instantaneous current available from the array, as well as the inherent 3

and automatic property controlling the output voltage of the array so that loads receive voltages within their own range of acceptability [16]. While batteries may seem like a good idea, they have a number of disadvantages. The type of lead-acid battery suitable for PV systems is a deep-cycle battery [15], which is different from one used for automobiles, and it is more expensive and not widely available. Battery lifetime in PV systems is typically three to eight years, but this reduces to typically two to six years in hot climate since high ambient temperature dramatically increases the rate of internal corrosion [24]. Batteries also require regular maintenance and will degrade very rapidly if the electrolyte is not topped up and the charge is not maintained. They reduce the efficiency of the overall system due to power loss during charge and discharge. Typical battery efficiency is around 85% but could go below 75% in hot climate [24]. From all those reasons, experienced PV system designers avoid batteries whenever possible. For water pumping systems, appropriately sized water reservoirs can meet the requirement of energy storage during the downtime of PV generation. The additional cost of reservoir is considerably lower than that incurred by the battery equipped system. As a matter of fact, only about five percent of solar pumping systems employ a battery bank [13]. 1.3 The Proposed System The experimental water pumping system proposed in this thesis is a stand-alone type without backup batteries. As shown in Figure 1-1, the system is very simple and consists of a single PV module, a maximum power point tracker (MPPT), and a DC water pump. The size of the system is inted to be small; therefore it could be built in the lab in the future. The system including the subsystems will be simulated to verify the functionalities. 4

PV Module [1] DC Water Pump [13] Figure 1-1: Block diagram of the proposed PV water pumping system 1.3.1 PV Module There are different sizes of PV module commercially available (typically sized from 60W to 170W). Usually, a number of PV modules are combined as an array to meet different energy demands. For example, a typical small-scale desalination plant requires a few thousand watts of power [24]. The size of system selected for the proposed system is 150W, which is commonly used in small water pumping systems for cattle grazing in rural areas of the United States. The power electronics lab located in the building 20, room 104, has three BP SX 150S multi-crystalline PV modules. Each module provides a maximum power of 150W [13], therefore the proposed system requires only one of them. A detailed discussion about PV and modeling of PV appears in Chapter 2. 1.3.2 Maximum Power Point Tracker The maximum power point tracker (MPPT) is now prevalent in grid-tied PV power systems and is becoming more popular in stand-alone systems. It should not be confused with sun trackers, mechanical devices that rotate and/or tilt PV modules in the direction of sun. MPPT is a power electronic device interconnecting a PV power source and a load, 5

maximizes the power output from a PV module or array with varying operating conditions, and therefore maximizes the system efficiency. MPPT is made up with a switch-mode DC- DC converter and a controller. For grid-tied systems, a switch-mode inverter sometimes fills the role of MPPT. Otherwise, it is combined with a DC-DC converter that performs the MPPT function. In addition to MPPT, the system could also employ a sun tracker. According to the data in reference [15], the single-axis sun tracker can collect about 40% more energy than a seasonally optimized fixed-axis collector in summer in a dry climate such as Albuquerque, New Mexico. In winter, however, it can gain only 20% more energy. In a climate with more water vapor in the atmosphere such as Seattle, Washington, the effect of sun tracker is smaller because a larger fraction of solar irradiation is diffuse. It collects 30% more energy in summer, but the gain is less than 10% in winter. The two-axis tracker is only a few percent better than the single-axis version. Sun tracking enables the system to meet energy demand with smaller PV modules, but it increases the cost and complexity of system. Since it is made of moving parts, there is also a higher chance of failure. Therefore, in this simple system, the sun tracker is not implemented. A detailed discussion on MPPT appears in Chapter 3. 1.3.3 MPPT Controller Analog controllers have traditionally performed control of MPPT. However, the use of digital controllers is rapidly increasing because they offer several advantages over analog controllers. First, digital controllers are programmable thus capable of implementing advanced algorithm with relative ease. It is far easier to code the equation, x = y z, than to design an analog circuit to do the same [23]. For the same reason, modification of the design 6

is much easier with digital controllers. They are immune to time and temperature drifts because they work in discrete, outside the linear operation. As a result, they offer long-term stability. They are also insensitive to component tolerances since they implement algorithm in software, where gains and parameters are consistent and reproducible [23]. They allow reduction of parts count since they can handle various tasks in a single chip. Many of them are also equipped with multiple A/D converters and PWM generators, thus they can control multiple devices with a single controller. This thesis, therefore, chooses a method of digital control for MPPT. The design and simulations of MPPT in Chapter 4 are done on the premise that it is going to be built with a microcontroller or a DSP, and the algorithm is readily transferable to its implementation. Chapter 3 provides discussions of various control methods. Appix B provides introduction of Texas Instruments DSP and SIMULINK as an implementation tool. 1.3.4 Water Pump Two types of pumps are commonly used for PV water pumping applications: positive displacement and centrifugal [19]. Positive displacement types are used in low-volume pumps [13] and cost-effective. Centrifugal pumps have relatively high efficiency [19] and are capable of pumping a high volume of water [13]. A typical size of system with this type pump is at least 500W or larger. There is a growing tr among the pump manufacturers to use them with brushless DC motors (BDCM) for higher efficiency and low maintenance [19]. However, the cost and complexity of these systems will be significantly higher. Water pumps are driven by various types of motors. AC induction motors are cheaper and widely available worldwide. The system, however, needs an inverter to convert DC output power from PV to AC power, which is usually expensive, and it is also less efficient than DC 7

motor-pump systems [19]. In general, DC motors are preferred because they are highly efficient and can be directly coupled with a PV module or array. Brushed types are less expensive and more common although brushes need to be replaced periodically (typically every two years) [19]. There is also an aforementioned brushless type. The water pump chosen here for its size and cost is the Kyocera SD 12-30 submersible solar pump, pictured in Figure 1-1. It is a diaphragm-type positive displacement pump equipped with a brushed permanent magnet DC motor and designed for use in standalone water delivery systems, specifically for water delivery in remote locations. Flow rates up to 17.0L/min (4.5GPM) and heads up to 30.0m (100ft.) [13]. The typical daily output is between 2,700L and 5,000L [13]. The rated maximum power consumption is 150W. It operates with a low voltage (12~30V DC), and its power requirement is as little as 35W [13]. A simple model of this water pump is used for simulations in Chapter 4. 1.4 Background and Scope of This Thesis The impetus for this research is to investigate the use of power electronics in renewable energy, particularly photovoltaics (PV). Numerous studies have been done in PV systems, a significant number of them in Europe, Japan, and Australia. In the United Sates, there is a growing interest in PV, but research and development in PV systems is far behind from the aforementioned countries, and unfortunately, California State Universities (CSUs) are no exception. There are only a small number of studies related to PV systems in the past. Among them, there were a few senior projects which built PV facilities here in California Polytechnic State University, San Luis Obispo. Two senior projects, also here, built a simple PV battery charger, and a few others dealt with a sun tracker. There have been only two master s theses written about PV systems in the CSU system. The first attempt to study 8

MPPT was made by Dang [4] of California Polytechnic State University, Pomona. The thesis built a small PV module simulator and a buck converter without a controller. Then, it provided a rudimentary computer simulation of MPPT with a resistive load. The study was, however, far from comprehensive. Another was done here by Day [5], and it centered round a power system for a miniature satellite. It included MPPT in the system, but the functionality of MPPT was not tested. The theoretical study was insufficient, and it lacked simulations and experiments to ensure the functionality of MPPT. MPPT is one of many applications of power electronics, and it is a relatively new and unknown area. There is no textbook that provides comprehensive and detailed explanations about MPPT. Therefore, this thesis investigates it in detail and provides better explanations for students who are interested in this research area. In order to understand and design MPPT, it is necessary to have a good understanding of the behaviors of PV. The thesis facilitates it using MATLAB models of PV cell and module. Each subsystem in the PV water pumping system is modeled for MATLAB simulations. Finally, the functionality of MPPT for water pumping systems is verified and validated. This thesis is limited to providing theoretical studies and simulations of PV water pumping system with MPPT. The system will not be built in this thesis; that is left as future work. Thus, it will not cover a discussion about actual implementation of DSP or microcontrollers, nor other hardware implementation, beyond a discussion on component selection for the DC-DC converter. A major assumption made in simulations is the use of an ideal DC-DC converter, as opposed to a more realistic model that includes losses. The model, however, should provide sufficient results for verification of MPPT functionality. 9

Chapter 2 Photovoltaic Modules 2.1 Introduction The history of PV dates back to 1839 when a French physicist, Edmund Becquerel, discovered the first photovoltaic effect when he illuminated a metal electrode in an electrolytic solution [16]. Thirty-seven years later British physicist, William Adams, with his student, Richard Day, discovered a photovoltaic material, selenium, and made solid cells with 1~2% efficiency which were soon widely adopted in the exposure meters of camera [16]. In 1954 the first generation of semiconductor silicon-based PV cells was born, with efficiency of 6% [3], and adopted in space applications. Today, the production of PV cells is following an exponential growth curve since technological advancement of late 80s that has started to rapidly improve efficiency and reduce cost. This chapter discusses the fundamentals of PV cells and modeling of a PV cell using an equivalent electrical circuit. The models are implemented using MATLAB to study PV characteristics and simulate a real PV module. 2.2 Photovoltaic Cell Photons of light with energy higher than the band-gap energy of PV material can make electrons in the material break free from atoms that hold them and create hole-electron pairs, as shown in Figure 2-1. These electrons, however, will soon fall back into holes causing charge carriers to disappear. If a nearby electric field is provided, those in the conduction band can be continuously swept away from holes toward a metallic contact where 10

they will emerge as an electric current. The electric field within the semiconductor itself at the junction between two regions of crystals of different type, called a p-n junction [16]. Figure 2-1: Illustration of the p-n junction of PV cell [16] Showing hole-electron pairs created by photons The PV cell has electrical contacts on its top and bottom to capture the electrons, as shown in Figure 2-2. When the PV cell delivers power to the load, the electrons flow out of the n-side into the connecting wire, through the load, and back to the p-side where they recombine with holes [16]. Note that conventional current flows in the opposite direction from electrons. Figure 2-2: Illustrated side view of solar cell and the conducting current [16] 11

2.3 Modeling a PV Cell The use of equivalent electric circuits makes it possible to model characteristics of a PV cell. The method used here is implemented in MATLAB programs for simulations. The same modeling technique is also applicable for modeling a PV module. 2.3.1 The Simplest Model The simplest model of a PV cell is shown as an equivalent circuit below that consists of an ideal current source in parallel with an ideal diode. The current source represents the current generated by photons (often denoted as I ph or I L ), and its output is constant under constant temperature and constant incident radiation of light. Figure 2-3: PV cell with a load and its simple equivalent circuit [16] There are two key parameters frequently used to characterize a PV cell. Shorting together the terminals of the cell, as shown in Figure 2-4 (a), the photon generated current will follow out of the cell as a short-circuit current (I sc ). Thus, I ph = I sc. As shown in Figure 2-4 (b), when there is no connection to the PV cell (open-circuit), the photon generated current is shunted internally by the intrinsic p-n junction diode. This gives the open circuit voltage (V oc ). The PV module or cell manufacturers usually provide the values of these parameters in their datasheets. 12

Figure 2-4: Diagrams showing a short-circuit and an open-circuit condition [16] The output current (I) from the PV cell is found by applying the Kirchoff s current law (KCL) on the equivalent circuit shown in Figure 2-3. I = I sc I d (2.1) where: I sc is the short-circuit current that is equal to the photon generated current, and I d is the current shunted through the intrinsic diode. The diode current I d is given by the Shockley s diode equation: where: I o is the reverse saturation current of diode (A), q is the electron charge (1.602 10-19 C), V d is the voltage across the diode (V), k is the Boltzmann s constant (1.381 10-23 J/K), T is the junction temperature in Kelvin (K). qvd / kt I = I ( e 1) (2.2) d o Replacing I d of the equation (2.1) by the equation (2.2) gives the current-voltage relationship of the PV cell. qv / kt I = I I ( e 1) (2.3) sc o where: V is the voltage across the PV cell, and I is the output current from the cell. 13

The reverse saturation current of diode (I o ) is constant under the constant temperature and found by setting the open-circuit condition as shown in Figure 2-4 (b). Using the equation (2.3), let I = 0 (no output current) and solve for I o. qvoc / kt 0 = I I ( e 1) (2.4) sc o qvoc / kt I = I ( e 1) (2.5) sc o I sc I o = qv / kt (2.6) oc ( e 1) To a very good approximation, the photon generated current, which is equal to I sc, is directly proportional to the irradiance, the intensity of illumination, to PV cell [15]. Thus, if the value, I sc, is known from the datasheet, under the standard test condition, G o =1000W/m 2 at the air mass (AM) = 1.5, then the photon generated current at any other irradiance, G (W/m 2 ), is given by: G I = I (2.7) sc G G o sc Figure 2-5 shows that current and voltage relationship (often called as an I-V curve) of an ideal PV cell simulated by MATLAB using the simplest equivalent circuit model. The discussion of MATLAB simulations will appear in Section 2.5. The PV cell output is both limited by the cell current and the cell voltage, and it can only produce a power with any combinations of current and voltage on the I-V curve. It also shows that the cell current is proportional to the irradiance. Go 14

5 4.5 Full Sun (1000W/m 2 ) 4 Cell Current (A) 3.5 3 2.5 2 1.5 Half Sun (500W/m 2 ) 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Cell Voltage (V) Figure 2-5: I-V plot of ideal PV cell under two different levels of irradiance (25 o C) 2.3.2 The More Accurate Model There are a few things that have not been taken into account in the simple model and that will affect the performance of a PV cell in practice. a) Series Resistance In a practical PV cell, there is a series of resistance in a current path through the semiconductor material, the metal grid, contacts, and current collecting bus [2]. These resistive losses are lumped together as a series resister (R s ). Its effect becomes very conspicuous in a PV module that consists of many series-connected cells, and the value of resistance is multiplied by the number of cells. b) Parallel Resistance This is also called shunt resistance. It is a loss associated with a small leakage of current through a resistive path in parallel with the intrinsic device [2]. This can be 15

16 represented by a parallel resister (R p ). Its effect is much less conspicuous in a PV module compared to the series resistance, and it will only become noticeable when a number of PV modules are connected in parallel for a larger system. c) Recombination Recombination in the depletion region of PV cells provides non-ohmic current paths in parallel with the intrinsic PV cell [2] [7]. As shown in Figure 2-6, this can be represented by the second diode (D2) in the equivalent circuit. Rp Rs D1 D2 Isc Load - + Figure 2-6: More accurate equivalent circuit of PV cell Summarizing these effects, the current-voltage relationship of PV cell is written as: + = + + p s kt R I V q o kt R I V q o sc R R I V e I e I I I s s 1 1 2 2 1 (2.8) It is possible to combine the first diode (D1) and the second diode (D2) and rewrite the equation (2.8) in the following form. + = + p s nkt R I V q o sc R R I V e I I I s 1 (2.9) where: n is known as the ideality factor ( n is sometimes denoted as A ) and takes the value between one and two [7]. V n=1 n=2

2.4 Photovoltaic Module A single PV cell produces an output voltage less than 1V, about 0.6V for crystallinesilicon (Si) cells, thus a number of PV cells are connected in series to archive a desired output voltage. When series-connected cells are placed in a frame, it is called as a module. Most of commercially available PV modules with crystalline-si cells have either 36 or 72 series-connected cells. A 36-cell module provides a voltage suitable for charging a 12V battery, and similarly a 72-cell module is appropriate for a 24V battery. This is because most of PV systems used to have backup batteries, however today many PV systems do not use batteries; for example, grid-tied systems. Furthermore, the advent of high efficiency DC-DC converters has alleviated the need for modules with specific voltages. When the PV cells are wired together in series, the current output is the same as the single cell, but the voltage output is the sum of each cell voltage, as shown in Figure 2-7. 5 4.5 4 3.5 Current (A) 3 2.5 2 1.5 3 cells 9 cells 36 cells 72 cells 1 0.5 0 0 5 10 15 20 25 30 35 40 45 0.6V for each cell Voltage (V) Figure 2-7: PV cells are connected in series to make up a PV module 17

Also, multiple modules can be wired together in series or parallel to deliver the voltage and current level needed. The group of modules is called an array. 2.5 Modeling a PV Module by MATLAB BP Solar BP SX 150S PV module, pictured in Figure 2-8, is chosen for a MATLAB simulation model. The module is made of 72 multi-crystalline silicon solar cells in series and provides 150W of nominal maximum power [1]. Table 2-1 shows its electrical specification. Figure 2-8: Picture of BP SX 150S PV module [1] Electrical Characteristics Maximum Power (P max ) 150W Voltage at P max (V mp ) 34.5V Current at P max (I mp ) 4.35A Open-circuit voltage (V oc ) 43.5V Short-circuit current (I sc ) 4.75A Temperature coefficient of I sc 0.065 ± 0.015 %/ o C Temperature coefficient of V oc -160 ± 20 mv/ o C Temperature coefficient of power -0.5 ± 0.05 %/ o C NOCT 47 ± 2 o C Table 2-1: Electrical characteristics data of PV module taken from the datasheet [1] The strategy of modeling a PV module is no different from modeling a PV cell. It uses the same PV cell model. The parameters are the all same, but only a voltage parameter (such as the open-circuit voltage) is different and must be divided by the number of cells. 18

The study done by Walker [27] of University of Queensland, Australia, uses the electric model with moderate complexity, shown in Figure 2-9, and provides fairly accurate results. The model consists of a current source (I sc ), a diode (D), and a series resistance (R s ). The effect of parallel resistance (R p ) is very small in a single module, thus the model does not include it. To make a better model, it also includes temperature effects on the short-circuit current (I sc ) and the reverse saturation current of diode (I o ). It uses a single diode with the diode ideality factor (n) set to achieve the best I-V curve match. Rs Isc D + V Load - Figure 2-9: Equivalent circuit used in the MATLAB simulations Since it does not include the effect of parallel resistance (R p ), letting R p = in the equation (2.9) gives the equation [27] that describes the current-voltage relationship of the PV cell, and it is shown below. V + I R s q nkt I = I sc I oe 1 (2.10) where: I is the cell current (the same as the module current), V is the cell voltage = {module voltage} {# of cells in series}, T is the cell temperature in Kelvin (K). 19

First, calculate the short-circuit current (I sc ) at a given cell temperature (T): I sc T sc T [ + a( T T )] = I 1 (2.11) ref where: I sc at T ref is given in the datasheet (measured under irradiance of 1000W/m 2 ), T ref is the reference temperature of PV cell in Kelvin (K), usually 298K (25 o C), a is the temperature coefficient of I sc in percent change per degree temperature also given in the datasheet. ref The short-circuit current (I sc ) is proportional to the intensity of irradiance, thus I sc at a given irradiance (G) is: G I = I (2.12) sc G G o sc where: G o is the nominal value of irradiance, which is normally 1KW/m 2. The reverse saturation current of diode (I o ) at the reference temperature (T ref ) is given by the equation (2.6) with the diode ideality factor added: I sc I o = qv / nkt (2.13) oc ( e 1) The reverse saturation current (I o ) is temperature depant and the I o at a given temperature (T) is calculated by the following equation [27]. Go I o 3 q Eg 1 1 n k T Tref n T T = Io T e (2.14) ref Tref The diode ideality factor (n) is unknown and must be estimated. It takes a value between one and two; the value of n=1 (for the ideal diode) is, however, used until the more accurate value is estimated later by curve fitting [27]. Figure 2-10 shows the effect of the varying ideality factor. 20

5 4.5 n=1 4 n=2 3.5 Module Current (A) 3 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 30 35 40 45 Module Voltage (V) Figure 2-10: Effect of diode ideally factors by MATLAB simulation (1KW/m 2, 25 o C) The series resistance (R s ) of the PV module has a large impact on the slope of the I-V curve near the open-circuit voltage (V oc ), as shown in Figure 2-11, hence the value of R s is dv calculated by evaluating the slope of the I-V curve at the Voc [27]. The equation for R s is di derived by differentiating the equation (2.10) and then rearranging it in terms of R s. V + I R s q nkt I = I sc I oe 1 (2.15) V + I R q nkt s dv + Rs di di = 0 I o q e (2.16) nkt di nkt q R s = V + I Rs dv q (2.17) nkt I e o 21

Then, evaluate the equation (2.17) at the open circuit voltage that is V=V oc (also let I=0). R s dv = di Voc I nkt q o e qvoc nkt (2.18) where: dv di V oc is the slope of the I-V curve at the V oc (use the I-V curve in the datasheet then divide it by the number of cells in series), V oc is the open-circuit voltage of cell (found by dividing V oc in the datasheet by the number of cells in series). The calculation using the slope measurement of the I-V curve published on the BP SX 150 datasheet gives a value of the series resistance per cell, R s = 5.1m. 5 4.5 4 3.5 Rs=0 Module Current (A) 3 2.5 2 1.5 1 Rs=5 mohm Rs=10 mohm Rs=15 mohm 0.5 0 0 5 10 15 20 25 30 35 40 45 Module Voltage (V) Figure 2-11: Effect of series resistances by MATLAB simulation (1KW/m 2, 25 o C) Finally, it is possible to solve the equation of I-V characteristics (2.10). It is, however, complex because the solution of current is recursive by inclusion of a series resistance in the 22

model. Although it may be possible to find the answer by simple iterations, the Newton s method is chosen for rapid convergence of the answer [27]. The Newton s method is described as: ( xn ) ( x ) where: f (x) is the derivative of the function, f ( x) = 0 next value. f xn+1 = xn (2.19) f n, n Rewriting the equation (2.10) gives the following function: x is a present value, and xn+ 1 is a V + I R s q nkt f ( I ) = I sc I I oe 1 = 0 (2.20) Plugging this into the equation (2.19) gives a following recursive equation, and the output current (I) is computed iteratively. I n+ V + I n Rs q nkt I sc I n I oe 1 = I (2.21) 1 n V + I n Rs q R q s nkt 1 I o e nkt The MATLAB function written in this thesis performs the calculation five times iteratively to ensure convergence of the results. The testing result has shown that the value of I n usually converges within three iterations and never more than four interactions. Please refer to Appix A.1.1 for this MATLAB function. Figure 2-12 shows the plots of I-V characteristics at various module temperatures simulated with the MATLAB model for BP SX 150S PV module. Data points superimposed on the plots are taken from the I-V curves published on the manufacturer s datasheet [1]. After some trials with various diode ideality factors, the MATLAB model chooses the value 23

of n = 1.62 that attains the best match with the I-V curve on the datasheet. The figure shows good correspondence between the data points and the simulated I-V curves. 75 0 C 25 0 C 50 0 C O 0 C Figure 2-12: I-V curves of BP SX 150S PV module at various temperatures Simulated with the MATLAB model (1KW/m 2, 25 o C) 24

2.6 The I-V Curve and Maximum Power Point Figure 2-13 shows the I-V curve of the BP SX 150S PV module simulated with the MATLAB model. A PV module can produce the power at a point, called an operating point, anywhere on the I-V curve. The coordinates of the operating point are the operating voltage and current. There is a unique point near the knee of the I-V curve, called a maximum power point (MPP), at which the module operates with the maximum efficiency and produces the maximum output power. It is possible to visualize the location of the by fitting the largest possible rectangle inside of the I-V curve, and its area equal to the output power which is a product of voltage and current. 5 Isc = 4.75A P3 = 94.9W P1 = 150.0W 4.5 Module Current (A) 4 3.5 3 2.5 2 1.5 1 Impp = 4.35A Maximum Power Point (MPP) P2 = 108.2W 0.5 Vmpp = 34.5V 0 0 5 10 15 20 25 30 35 40 45 50 Module Voltage (V) Voc = 43.5V Figure 2-13: Simulated I-V curve of BP SX 150S PV module (1KW/m 2, 25 o C) The power vs. voltage plot is overlaid on the I-V plot of the PV module, as shown in Figure 2-14. It reveals that the amount of power produced by the PV module varies greatly 25

deping on its operating condition. It is important to operate the system at the MPP of PV module in order to exploit the maximum power from the module. The next chapter will discuss how to do it. 8 160 7 Pmax 140 6 120 Module Current (A) 5 4 3 2 Isc Impp MPP 100 80 60 40 Module Output Power (W) 1 Vmpp Voc 20 0 0 5 10 15 20 25 30 35 40 45 0 Module Voltage (V) Figure 2-14: I-V and P-V relationships of BP SX 150S PV module Simulated with the MATLAB model (1KW/m 2, 25 o C) 26

Chapter 3 Maximum Power Point Tracker 3.1 Introduction When a PV module is directly coupled to a load, the PV module s operating point will be at the intersection of its I V curve and the load line which is the I-V relationship of load. For example in Figure 3-1, a resistive load has a straight line with a slope of 1/R load as shown in Figure 3-2. In other words, the impedance of load dictates the operating condition of the PV module. In general, this operating point is seldom at the PV module s MPP, thus it is not producing the maximum power. A study shows that a direct-coupled system utilizes a mere 31% of the PV capacity [11]. A PV array is usually oversized to compensate for a low power yield during winter months. This mismatching between a PV module and a load requires further over-sizing of the PV array and thus increases the overall system cost. To mitigate this problem, a maximum power point tracker (MPPT) can be used to maintain the PV module s operating point at the MPP. MPPTs can extract more than 97% of the PV power when properly optimized [9]. This chapter discusses the I-V characteristics of PV modules and loads, matching between the two, and the use of DC-DC converters as a means of MPPT. It also discusses the details of some MPPT algorithms and control methods, and limitations of MPPT. + I PV V R - Figure 3-1: PV module is directly connected to a (variable) resistive load 27

5 4.5 4 R=4 Ohms * R=7.93 Ohms * MPP 3.5 Module Current (A) 3 2.5 2 Slope=1/R R=16 Ohms * 1.5 1 0.5 Increasing R 0 0 5 10 15 20 25 30 35 40 45 50 Module Voltage (V) Figure 3-2: I-V curves of BP SX 150S PV module and various resistive loads Simulated with the MATLAB model (1KW/m 2, 25 o C) 3.2 I-V Characteristics of DC Motors Many PV water pumping systems employ DC motors (instead of AC motors) because they could be directly coupled with PV arrays and make a very simple system. Among different types of DC motors, a permanent magnet DC (PMDC) motor is preferred in PV systems because it can provide higher starting torque. Figure 3-3 shows an electrical model of a PMDC motor. When the motor is turning, it produces a back emf, or a counterelectromotive force, described as an electric potential (E) proportional to the angular speed () of the rotor. From the equivalent circuit, the DC voltage equation for the armature circuit is: V = I Ra + K ω (3.1) where: R a is the armature resistance. 28

The back emf is E=K where: K is the constant, and is the angular speed of rotor in rad/sec. Ra PV + V I E=Kw - Figure 3-3: Electrical model of permanent magnet DC motor Figure 3-4 shows an example of current-voltage relationship (I-V curve) of a DC motor. Applying the voltage to start the motor, the current rises rapidly with increasing voltage until the current is sufficient to create enough starting torque to break the motor loose from static friction [16]. At start-up (=0), there is no effect of back emf, therefore the starting current builds up linearly with a steep slope of 1/R a on the I-V plot as shown in Figure 3-4. Once it starts to run, the back emf takes effect and drops the current, therefore the current rises slowly with increasing voltage. As mentioned already a simple type of PV water pumping systems uses a direct coupled PV-motor setup. This configuration has a severe disadvantage in efficiency because of a mismatched operating point, as shown in Figure 3-4. For this example, the water pumping system would not start operating until irradiance reaches at 400W/m 2. Once it starts to run, it requires as little as 200W/m 2 of irradiance to maintain the minimum operation. This means that the system cannot utilize a fair amount of morning insolation just because there is insufficient starting torque. Also, when the motor is operated under the locked condition for 29

a long time, it may result in shortening of the life of the motor due to input electrical energy converted to heat rather than to mechanical output [15]. 1000W/m 2 DC Motor I-V Curve Slope = 1/R a 800W/m 2 Current 600W/m 2 400W/m 2 200W/m 2 Voltage Figure 3-4: PV I-V curves with varying irradiance and a DC motor I-V curve There is a MPPT specifically called a linear current booster (LCB) that is designed to overcome the above mentioned problem in water pumping systems. The MPPT maintains the input voltage and current of LCB at the MPP of PV module. As shown in Figure 3-5, the power produced at the MPP is relatively low-current and high-voltage which is opposite of those required by the pump motor. The LCB shifts this relationship around and converts into high-current and low-voltage power which satisfies the pump motor characteristics. For the example in Figure 3-5, tracing of the iso-power (constant power) line from the MPP reveals that the LCB could start the pump motor with as little as 50W/m 2 of irradiance (assuming the LCB can convert the power without loss). 30

DC Motor I-V Curve 1000W/m 2 800W/m 2 MPP Iso-power line 600W/m 2 Current 400W/m 2 200W/m 2 50W/m 2 Voltage Figure 3-5: PV I-V curves with iso-power lines (dotted) and a DC motor I-V curve 3.3 DC-DC Converter The heart of MPPT hardware is a switch-mode DC-DC converter. It is widely used in DC power supplies and DC motor drives for the purpose of converting unregulated DC input into a controlled DC output at a desired voltage level [17]. MPPT uses the same converter for a different purpose: regulating the input voltage at the PV MPP and providing loadmatching for the maximum power transfer. 3.3.1 Topologies There are a number of different topologies for DC-DC converters. They are categorized into isolated or non-isolated topologies. The isolated topologies use a small-sized high-frequency electrical isolation transformer which provides the benefits of DC isolation between input and output, and step 31

up or down of output voltage by changing the transformer turns ratio. They are very often used in switch-mode DC power supplies [18]. Popular topologies for a majority of the applications are flyback, half-bridge, and full-bridge [22]. In PV applications, the grid-tied systems often use these types of topologies when electrical isolation is preferred for safety reasons. Non-isolated topologies do not have isolation transformers. They are almost always used in DC motor drives [17]. These topologies are further categorized into three types: step down (buck), step up (boost), and step up & down (buck-boost). The buck topology is used for voltage step-down. In PV applications, the buck type converter is usually used for charging batteries and in LCB for water pumping systems. The boost topology is used for stepping up the voltage. The grid-tied systems use a boost type converter to step up the output voltage to the utility level before the inverter stage. Then, there are topologies able to step up and down the voltage such as: buck-boost, Cúk, and SEPIC (stands for Single Ended Primary Inductor Converter). For PV system with batteries, the MPP of commercial PV module is set above the charging voltage of batteries for most combinations of irradiance and temperature. A buck converter can operate at the MPP under most conditions, but it cannot do so when the MPP goes below the battery charging voltage under a low-irradiance and high-temperature condition. Thus, the additional boost capability can slightly increase the overall efficiency [27]. 3.3.2 Cúk and SEPIC Converters For water pumping systems, the output voltage needs to be stepped down to provide a higher starting current for a pump motor. The buck converter is the simplest topology and easiest to understand and design, however it exhibits the most severe destructive failure mode 32

of all configurations [22]. Another disadvantage is that the input current is discontinuous because of the switch located at the input, thus good input filter design is essential. Other topologies capable of voltage step-down are Cúk and SEPIC. Even though their voltage step-up function is optional for LCB application, they have several advantages over the buck converter. They provide capacitive isolation which protects against switch failure (unlike the buck topology) [21]. The input current of the Cúk and SEPIC topologies is continuous, and they can draw a ripple free current from a PV array that is important for efficient MPPT. Figure 3-6 shows a circuit diagram of the basic Cúk converter. It is named after its inventor. It can provide the output voltage that is higher or lower than the input voltage. The SEPIC, a derivative of the Cúk converter, is also able to step up and down the voltage. Figure 3-7 shows a circuit diagram of the basic SEPIC converter. The characteristics of two topologies are very similar. They both use a capacitor as the main energy storage. As a result, the input current is continuous. The circuits have low switching losses and high efficiency [18]. The main difference is that the Cúk converter has a polarity of the output voltage reverse to the input voltage. The input and output of SEPIC converter have the same voltage polarity; therefore the SEPIC topology is sometimes preferred to the Cúk topology. SEPIC maybe also preferred for battery charging systems because the diode placed on the output stage works as a blocking diode preventing an adverse current going to PV source from the battery. The same diode, however, gives the disadvantage of high-ripple output current. On the other hand, the Cúk converter can provide a better output current characteristic due to the inductor on the output stage. Therefore, the thesis decides on the Cúk converter because of the good input and output current characteristics. 33

Figure 3-6: Circuit diagram of the basic Cúk converter Figure 3-7: Circuit diagram of the basic SEPIC converter 3.3.3 Basic Operation of Cúk Converter The basic operation of Cúk converter in continuous conduction mode is explained here. In steady state, the average inductor voltages are zero, thus by applying Kirchoff s voltage law (KVL) around outermost loop of the circuit shown in Figure 3-6 [21]. V = V + V C1 s o (3.2) Assume the capacitor (C 1 ) is large enough and its voltage is ripple free even though it stores and transfer large amount of energy from input to output [17] (this requires a good low ESR capacitor [21]). The initial condition is when the input voltage is turned on and switch (SW) is off. The diode (D) is forward biased, and the capacitor (C 1 ) is being charged. The operation of circuit can be divided into two modes. 34

Mode 1: When SW turns ON, the circuit becomes one shown in Figure 3-8. Figure 3-8: Basic Cúk converter when the switch is ON The voltage of the capacitor (C 1 ) makes the diode (D) reverse-biased and turned off. The capacitor (C 1 ) discharge its energy to the load through the loop formed with SW, C 2, R load, and L 2. The inductors are large enough, so assume that their currents are ripple free. Thus, the following relationship is established [21]. I = I (3.3) C1 L2 Mode 2: When SW turns OFF, the circuit becomes one shown in Figure 3-9. Figure 3-9: Basic Cúk converter when the switch is OFF The capacitor (C 1 ) is getting charged by the input (V s ) through the inductor (L 1 ). The energy stored in the inductor (L 2 ) is transfer to the load through the loop formed by D, C 2, and R load. Thus, the following relationship is established [21]. I = I (3.4) C1 L1 35

For periodic operation, the average capacitor current is zero. Thus, from the equation (3.3) and (3.4) [21]: [ I 1 ] DT + [ I C1 ] (1 D) T = 0 C (3.5) SW ON SW OFF I L 2 DT + I L1 (1 D) T = 0 (3.6) I I L1 L2 D = 1 D (3.7) where: D is the duty cycle (0 < D < 1), and T is the switching period. Assuming that this is an ideal converter, the average power supplied by the source must be the same as the average power absorbed by the load [21]. P in = P out (3.8) V s I = V I (3.9) L1 o L2 I I L1 L2 V = V o s (3.10) Combining the equation (3.7) and (3.10), the following voltage transfer function is derived [21]. V V o s D = (3.11) 1 D Its relationship to the duty cycle (D) is: If 0 < D < 0.5 the output is smaller than the input. If D = 0.5 the output is the same as the input. If 0.5 < D < 1 the output is larger than the input. 36

3.4 Mechanism of Load Matching As described in Section 3.1, when PV is directly coupled with a load, the operating point of PV is dictated by the load (or impedance to be specific). The impedance of load is described as below. V o R load = (3.12) I o where: V o is the output voltage, and I o is the output current. The optimal load for PV is described as: V MPP R opt = (3.13) I MPP where: V MPP and I MPP are the voltage and current at the MPP respectively. When the value of R load matches with that of R opt, the maximum power transfer from PV to the load will occur. These two are, however, indepent and rarely matches in practice. The goal of the MPPT is to match the impedance of load to the optimal impedance of PV. The following is an example of load matching using an ideal (loss-less) Cúk converter. From the equation (3.11): From the equation (3.10), V D = 1 (3.14) D s V o I I s o = I I L1 L2 V = V o s (3.15) From the equation (3.14) and (3.15), I D = (3.16) 1 D s I o 37

From the equation (3.14) and (3.16), the input impedance of the converter is: R in V (1 D) V (1 D) 2 = s o = = 2 I s D I o D 2 2 R load (3.17) As shown in Figure 3-10, the impedance seem by PV is the input impedance of the converter (R in ). By changing the duty cycle (D), the value of R in can be matched with that of R opt. Therefore, the impedance of the load can be anything as long as the duty cycle is adjusted accordingly. + PV Rin DC-DC Conv Rload - Figure 3-10: The impedance seen by PV is R in that is adjustable by duty cycle (D) 3.5 Maximum Power Point Tracking Algorithms The location of the MPP in the I V plane is not known beforehand and always changes dynamically deping on irradiance and temperature. For example, Figure 3-11 shows a set of PV I V curves under increasing irradiance at the constant temperature (25 o C), and Figure 3-12 shows the I V curves at the same irradiance values but with a higher temperature (50 o C). There are observable voltage shifts where the MPP occurs. Therefore, the MPP needs to be located by tracking algorithm, which is the heart of MPPT controller. There are a number of methods that have been proposed. One method measures an open-circuit voltage (V oc ) of PV module every 30 seconds by disconnecting it from rest of the circuit for a short moment. Then, after re-connection, the module voltage is adjusted to 76% 38

of measured V oc which corresponds to the voltage at the MPP [6] (note: the percentage deps on the type of cell used). The implementation of this open-loop control method is very simple and low-cost although the MPPT efficiencies are relatively low (between 73~91%) [9]. Model calculations can also predict the location of MPP; however in practice it does not work well because it does not take physical variations and aging of module and other effects such as shading into account. Furthermore, a pyranometer that measures irradiance is quite expensive. Search algorithm using a closed-loop control can achieve higher efficiencies, thus it is the customary choice for MPPT. Among different algorithms, the Perturb & Observe (P&O) and Incremental Conductance (inccond) methods are studied here. 5 4.5 4 3.5 1000W/m 2 750W/m 2 Maximum Power Point Module Current (A) 3 2.5 2 1.5 1 500W/m 2 250W/m 2 0.5 50W/m 2 0 0 5 10 15 20 25 30 35 40 45 50 Module Voltage (V) Figure 3-11: I-V curves for varying irradiance and a trace of MPPs (25 o C) 39

5 4.5 4 3.5 1000W/m 2 750W/m 2 Maximum Power Point Module Current (A) 3 2.5 2 1.5 1 500W/m 2 250W/m 2 0.5 50W/m 2 0 0 5 10 15 20 25 30 35 40 45 50 Module Voltage (V) Figure 3-12: I-V curves for varying irradiance and a trace of MPPs (50 o C) 3.5.1 Perturb & Observe Algorithm The perturb & observe (P&O) algorithm, also known as the hill climbing method, is very popular and the most commonly used in practice because of its simplicity in algorithm and the ease of implementation. The most basic form of the P&O algorithm operates as follows. Figure 3-13 shows a PV module s output power curve as a function of voltage (P-V curve), at the constant irradiance and the constant module temperature, assuming the PV module is operating at a point which is away from the MPP. In this algorithm the operating voltage of the PV module is perturbed by a small increment, and the resulting change of power, P, is observed. If the P is positive, then it is supposed that it has moved the operating point closer to the MPP. Thus, further voltage perturbations in the same direction should move the operating point toward the MPP. If the P is negative, the 40

operating point has moved away from the MPP, and the direction of perturbation should be reversed to move back toward the MPP. Figure 3-14 shows the flowchart of this algorithm. 160 140 MPP 120 Module Output Power (W) 100 80 60 40 A * * B 20 0 0 5 10 15 20 25 30 35 40 45 50 Module Voltage (V) Figure 3-13: Plot of power vs. voltage for BP SX 150S PV module (1KW/m 2, 25 o C) Figure 3-14: Flowchart of the P&O algorithm 41

There are some limitations that reduce its MPPT efficiency. First, it cannot determine when it has actually reached the MPP. Instead, it oscillates the operating point around the MPP after each cycle and slightly reduces PV efficiency under the constant irradiance condition [9]. Second, it has been shown that it can exhibit erratic behavior in cases of rapidly changing atmospheric conditions as a result of moving clouds [11]. The cause of this problem can be explained using Figure 3-15 with a set of P-V curves with varying irradiance. Assume that the operating point is initially at the point A and is oscillating around the MPP at the irradiance of 250W/m 2. Then, the irradiance increases rapidly to 500W/m 2. The power measurement results in a positive P. If this operating point is perturbing from right to left around the MPP, then the operating point will actually moves from the point A toward the point E (instead of B). This happens because the MPPT can not tell that the positive P is the result of increasing irradiation and simply assumes that it is the result of moving the operating point to closer to the MPP. In this case the positive P is measured when the operating voltage has been moving toward the left; the MPPT is fooled as if there is a MPP on the left side. If the irradiance is still rapidly increasing, again the MPPT will see the positive P and will assume it is moving towards the MPP, continuing to perturb to the left. From points A, E, F and G, the operating point continues to deviate from the actual MPP until the solar radiation change slows or settles down. This situation can occur on partly cloudy days, and MPP tracking is most difficult because of the frequent movement of the MPP. 42

1000W/m 2 D Module Output Power (W) * G F * E * 750W/m 2 C 500W/m 2 B 250W/m 2 A Module Voltage (V) Figure 3-15: Erratic behavior of the P&O algorithm under rapidly increasing irradiance The advent of digital controller made implementation of algorithm easy; as a result many variations of the P&O algorithm were proposed to claim improvements. The problem of oscillations around the MPP can be solved by the simplest way of making a bypass loop which skips the perturbation when the P is very small which occurs near the MPP. The tradeoffs are a steady state error and a high risk of not detecting a small power change. Another way is the addition of a waiting function that causes a momentary cessation of perturbations if the direction of the perturbation is reversed several times in a row, indicating that the MPP has been reached [9]. It works well under the constant irradiation but makes the MPPT slower to respond to changing atmospheric conditions. A more complex one uses a variable step size of perturbation, using the slope of PV power as a variable, for example: V ref P, new = Vref + C [4] [12]. Again, this works well under the constant irradiation but V worsens the erratic behavior under rapidly changing atmospheric conditions on partly cloudy 43

days because the power change due to irradiance makes the step size too big. A modification involving taking a PV power measurement twice at the same voltage solves the problem of not detecting the changing irradiance [9]. Comparing these two measurements, the algorithm can determine whether the irradiance is changing and decide how to perturb the operating point. The tradeoff is that the increased number of sampling slows response times and increases the complexity of algorithm. 3.5.2 Incremental Conductance Algorithm In 1993 Hussein, Muta, Hoshino, and Osakada of Saga University, Japan, proposed the incremental conductance (inccond) algorithm inting to solve the problem of the P&O algorithm under rapidly changing atmospheric conditions [11]. The basic idea is that the slope of P-V curve becomes zero at the MPP, as shown in Figure 3-13. It is also possible to find a relative location of the operating point to the MPP by looking at the slopes. The slope is the derivative of the PV module s power with respect to its voltage and has the following relationships with the MPP. dp dv = 0 at MPP (3.18) dp dv dp dv > 0 < 0 at the left of MPP (3.19) at the right of MPP (3.20) The above equations are written in terms of voltage and current as follows. dp dv = d( V I) dv = dv I dv + V di dv = I + V di dv (3.21) 44

If the operating point is at the MPP, the equation (3.21) becomes: di I + V = 0 (3.22) dv di dv I = (3.23) V If the operating point is at the left side of the MPP, the equation (3.21) becomes: di I + V > 0 (3.24) dv di dv I > (3.25) V If the operating point is at the right side of the MPP, the equation (3.21) becomes: di I + V < 0 (3.26) dv di dv I < (3.27) V Note that the left side of the equations (3.23), (3.25), and (3.27) represents incremental conductance of the PV module, and the right side of the equations represents its instantaneous conductance. The flowchart shown in Figure 3-16 explains the operation of this algorithm. It starts with measuring the present values of PV module voltage and current. Then, it calculates the incremental changes, di and dv, using the present values and previous values of voltage and current. The main check is carried out using the relationships in the equations (3.23), (3.25), and (3.27). If the condition satisfies the inequality (3.25), it is assumed that the operating point is at the left side of the MPP thus must be moved to the right by increasing the module voltage. Similarly, if the condition satisfies the inequality (3.27), it is assumed that the operating point is at the right side of the MPP, thus must be moved to the left by decreasing 45

the module voltage. When the operating point reaches at the MPP, the condition satisfies the equation (3.23), and the algorithm bypasses the voltage adjustment. At the of cycle, it updates the history by storing the voltage and current data that will be used as previous values in the next cycle. Another important check included in this algorithm is to detect atmospheric conditions. If the MPPT is still operating at the MPP (condition: dv = 0) and the irradiation has not changed (condition: di = 0), it takes no action. If the irradiation has increased (condition: di > 0), it raises the MPP voltage. Then, the algorithm will increase the operating voltage to track the MPP. Similarly, if the irradiation has decreased (condition: di < 0), it lowers the MPP voltage. Then, the algorithm will decrease the operating voltage. di dv = I V di dv > I V Figure 3-16: Flowchart of the inccond algorithm 46

In practice, the condition dp/dv = 0 (or di/dv = -I/V) seldom occurs because of the approximation made in the calculation of di and dv [11]. Thus, a small margin of error (E) should be allowed, for example: dp/dv = ±E. The value of E is optimized with exchange between an amount of the steady-sate tracking error and a risk of oscillation of the operating point. 3.6 Control of MPPT As explained in the previous section, the MPPT algorithm tells a MPPT controller how to move the operating voltage. Then, it is a MPPT controller s task to bring the voltage to a desired level and maintain it. There are several methods often used for MPPT. 3.6.1 PI Control As shown in Figure 3-17, the MPPT takes measurement of PV voltage and current, and then tracking algorithm (P&O, inccond, or variations of two) calculates the reference voltage (V ref ) where the PV operating voltage should move next. The task of MPPT algorithm is to set V ref only, and it is repeated periodically with a slower rate (typically 1~10 samples per second). Then, there is another control loop that the proportional and integral (PI) controller regulates the input voltage of converter. Its task is to minimize error between V ref and the measured voltage by adjusting the duty cycle. The PI loop operates with a much faster rate and provides fast response and overall system stability [10] [12]. The PI controller itself can be implemented with analog components, but it is often done with DSP-based controller [10] because the DSP can handle other tasks such as MPP tracking thus reducing parts count. 47

Figure 3-17: Block diagram of MPPT with the PI compensator 3.6.2 Direct Control As shown in Figure 3-18, this control method is simpler and uses only one control loop, and it performs the adjustment of duty cycle within the MPP tracking algorithm. The way how to adjust the duty cycle is totally based on the theory of load matching explained in Section 3.4. Figure 3-18: Block diagram of MPPT with the direct control 48